IMO 1986 LL LUX50

Origin: LUX

Problem

Let D be the point on the side BC of the triangle ABC such that AD is the bisector of \angleCAB. Let I be the incenter of \triangleABC. (a) Construct the points P and Q on the sides AB and AC, respectively, such that PQ is parallel to BC and the perimeter of the triangle APQ is equal to k \cdot BC, where k is a given rational number. (b) Let R be the intersection point of PQ and AD. For what value of k does the equality AR = RI hold? (c) In which case do the equalities AR = RI = ID hold?